Low-Velocity Impact Behavior of Sandwich Plates with FG-CNTRC Face Sheets and Negative Poisson’s Ratio Auxetic Honeycombs Core

The combination of auxetic honeycomb and CNT reinforcement composite is expected to further improve the impact protection performance of sandwich structures. This paper studies the low-velocity impact response of sandwich plates with functionally graded carbon nanotubes reinforced composite (FG-CNTRC) face sheets and negative Poisson’s ratio (NPR) auxetic honeycomb core. The material properties of FG-CNTRC were obtained by the rule of mixture theory. The auxetic honeycomb core is made of Ti-6Al-4V. The governing equations are derived based on the first-order shear deformation theory and Hamilton’s principle. The nonlinear Hertz contact law is used to calculate the impact parameters. The Ritz method with Newmark’s time integration schemes is used to solve the response of the sandwich plates. The (20/−20/20)s, (45/−45/45)s and (70/−70/70)s stacking sequences of FG-CNTRC are considered. The effects of the gradient forms of FG-CNTRC surfaces, volume fractions of CNTs, impact velocities, temperatures, ratio of plate length, width and thickness of surface layers on the value of the plate center displacement, the recovery time of deformation, contact force and contact time of low-velocity impact were analyzed in detail.

Most natural materials have the properties of expanding (contracting) laterally when compressed (stretched) longitudinally, which can be defined as positive Poisson's ratio materials. In recent years, auxetic material has generated a lot of interest among researchers due to the negative Poisson's ratio (NPR) properties [43][44][45]. Re-entrant [46], chiral [47] and other various materials have been proposed. Due to the outstanding performance on energy absorption [48][49][50], crashworthiness [51,52], and low-velocity impact resistance [53,54], auxetic material has been increasingly applied in biological medicine, photonics, energy storage, thermal management, and acoustic areas [55]. As an ideal core of sandwich structures, auxetic material could be used in shield structures in aerospace and civil engineering. Therefore, the nonlinear mechanical response of the sandwich structure with an auxetic honeycomb core [56,57] was analyzed by Li, Shen, and Wang [58][59][60][61][62][63][64]. Wan et al. [65] analyzed the uniaxial compression or expanded properties of auxetic honeycombs. Grima et al. [66] proposed a hexagonal honeycomb with zero Poisson's ratios. Assidi and Ganghoffer [67] represented a composite with auxetic behavior and proved that the overall NPR could improve the mechanical properties. Grujicic et al. [68] focused on the sandwich structures with an auxetic hexagonal core and built the multi-physics model of fabrication and dynamic performance. Liu et al. [69] investigated the propagation of waves in a sandwich plate with a periodic composite core. Qiao and Chen [70] analyzed the impact response of auxetic double arrowhead honeycombs. Zhang et al. [71] analyzed the in-plane dynamic crushing behaviors and energy-absorbed characteristics of NPR honeycombs with cell microstructure. Zhang et al. [72] analyzed the dynamic mechanical and impact response on yarns with helical auxetic properties.
There are two main methods to propose auxetic structures: the first is using auxetic material as the core of sandwich plate [55]; and the second is changing the stacking sequence and orientation of laminate [73,74]. To realize a larger NPR value using the second method requires not only a specific stacking sequence but also a highly anisotropic properties of each ply [75]. Due to the mechanical properties of CNTs, the longitudinal elastic modulus E 11 of CNTRC is much larger than the transverse elastic modulus E 22 and large NPR properties can be proposed by designing the stacking sequence of CNTRC laminate. Then, Shen et al. [45,76] introduced the NPR property to the FG-CNTRC laminate and analyzed the nonlinear bending and free vibration response. Yang, Huang, and Shen [77,78], as well as Yu and Shen [79] analyzed the effects of an out-of-plane NPR property on large amplitude vibration and nonlinear bending of the FG-CNTRC laminated beam and plate. Fan, Wang [80] and Huang et al. [81,82] analyzed the dynamic response of the auxetic FG-CNTRC.
The combination of auxetic honeycomb and CNT reinforcement composite is expected to further improve the impact protection performance of sandwich structures. This paper studies the low-velocity impact response of the sandwich plates with functionally graded carbon nanotubes reinforced composite (FG-CNTRC) face sheets and a negative Poisson's ratio (NPR) auxetic honeycomb core. The rule of mixture theory was used to calculate the material properties of FG-CNTRC with the PmPV matrix and CNTs reinforcement, while the effective Poisson's ratio was obtained by laminate plate theory (Section 2.2). The NPR honeycomb core was made of Ti-6Al-4V (Section 2.3). The first-order shear deformation theory and Hamilton's principle were used to describe the governing equations of the plate (Section 3.1). The nonlinear Hertz contact law was used to calculate the impact parameters (Section 3.2). The Ritz method with Newmark's time integration schemes was used to solve the response of the sandwich plate (Section 3.3). After verifying the model, the (20/−20/20)s, (45/−45/45)s and (70/−70/70)s three kinds of stacking sequence of FG-CNTRC surfaces were considered. The effects of gradient forms of FG-CNTRC surfaces, volume fractions of CNTs, impact velocities, temperatures, ratio of plate length and the width and thickness of surface layers on low-velocity impact response were analyzed. The value of plate center displacement, recovery time of deformation, contact force and contact time were discussed in detail.

Modeling of Sandwich Plates
The sandwich plates with length a, width b and total thickness h are considered in this research, as shown in Figure 1. The face sheets with a thickness h f are FG-CNTRClaminated structures composed of CNTRC layers with various volume fractions of CNTs. The auxetic core with a thickness of h c is the negative Poisson's ratio honeycomb structure using isotropic titanium alloy (Ti-6Al-4V). A coordinate system (x, y, z) with (x, y) plane in the middle surface of the plate and z in the thickness direction is considered.

Materials of FG-CNTRC Face Sheets
The CNTRC layers with the poly(m-phenylenevinylene)-co-((2,5-dioctoxy-p-phenylene) vinylene) (PmPV) matrix are considered in this research. The material properties of the face sheets can be obtained based on the rule of mixture theory [4].
where the superscript c and m represent the material properties of CNTs and the matrix, respectively. V is the volume fraction, in which V m + V c = 1. η j (j = 1, 2, 3) is the efficiency parameters of CNTs. The values are shown in Table 1. E, G, ν, ρ and α are the elastic module, shear module, Poisson's ratio, density and the thermal expansion of the materials, respectively. The (10, 10) SWCNTs are considered as the reinforcement in this research and the material properties are shown in Table 2. The material properties of the matrix PmPV are shown in Table 3.  The functionally graded properties of the CNTRC laminated structure are established according to the arrangement of CNTRC layers with the CNTs' volume fractions of 0.11, 0.14 and 0.17. As shown in Figure 2 0.14 0.14 0.14 0.14 0.14 0.14 For an anisotropic laminated plate, the effective Poisson's ratios ν e 13 and ν e 23 can be expressed as [44] ν e 13 = − where A, B and D are the stiffness matrix of the FG-CNTRC laminated surface. The aforementioned elements of the matrix are presented in Appendix A.
Combining the gradient forms of FG-CNTRC, the effective Poisson's ratios could be calculated as shown in

Materials of Auxetic Honeycomb Core
The honeycomb core made of Ti-6Al-4V with negative Poisson's ratio properties is considered in this research. The unit cell of the honeycomb is shown in Figure 4 and the material properties of the honeycomb core can be obtained by [56] where the superscript h and subscript Ti represent the material properties of honeycomb and Ti-6Al-4V, respectively. l h represents the length of the inclined cell rib; t h represents the thickness of the cell rib; h h represents the length of the vertical cell rib; and θ h represents the inclined angle. The original properties of the honeycomb can be controlled by the parameters above. The material properties of the Ti-6Al-4V are mentioned in Table 4.

Governing Equations
The first-order shear deformation theory is used to describe the sandwich plate with length a, width b and thickness h, as shown in Figure 1 where u, v and w are the translation displacement components at the mid-plane in the x, y and z directions, respectively. φ x and φ y denote the rotation of the normal to the mid-plane along the y axis and x axis, respectively. The relationship between strain and displacement can be expressed as Considering the temperature effect, the stress component based on a linear constitutive relationship can be written as where ∆T is the temperature change and the transformed stiffnessQ can be calculated by where s and c are the sin and cos of the lamination angle against the x axis of the plate. Furthermore, the stiffness parameters can be given as The strain energy of the sandwich plate U p can be expressed as whereε = (ε 0 , κ 0 , γ 0 ) T is the strain matrix, S is the material constant matrix and where A, B, D, A s are the matrices of the plate stiffness, which can be calculated by where the transverse shear correction coefficient K s can be calculated by , f unctionally graded material (14) where ν and V are the Poisson's ratios and volume fraction of each material in the entire cross-section. The kinetic energy of the sandwich plate T can be obtained by The external virtual work δW can be obtained by where F c (t) is the contact force between the plate and the impactor, and µ is the deflection of the sandwich plate. Then, the total energy function based on Hamilton's principle can be expressed as The boundary conditions for the clamped of the plate edge can be expressed as

Low-Velocity Impact Response
Based on the nonlinear Hertz contact law, the contact force F c (t) between the sandwich plate and a steel ball can be obtained by [83] where µ = w i − w p is the deflection of the sandwich plate, and w i , w p refers to the displacement of the impactor and plate center, respectively. The subscript m refers to the maximum value of the variables. K c is the contact coefficient, which can be expressed as [83], where E i , ν i , r i are the elasticity modulus, Poisson's ratios and the radius of the impactor, respectively. E 2 is the transverse elasticity modulus of the sandwich plate. The displacement of the impactor w i can be calculated by where v i and m i are the velocity and mass of the impactor, respectively. Then, the Equation (19) can be obtained by

Solution Procedure
The Ritz method is considered to deduce the governing equations of motion from the total energy function in the spatial domain, and the functions of the displacement field can be expressed as where p n (x, y) are the shape functions. n = 1, 2, · · · , N and N is the number of terms in the basis. U n (t), V n (t), W n (t), Φ xn (t), Φ yn (t) are the unknown coefficients chosen according to the boundary conditions. The shape functions of the polynomial are considered in this research [84,85].
The equations of motion of the sandwich plate and impactor can be obtained by where q, M, K, F are the degrees of the freedom vector, mass matrix, stiffness matrix and impact load vector, respectively. Furthermore, the components of the mass matrix and the stiffness matrix are presented in Appendix B. The dot over the variable refers to the differentiation of that variable with respect to time. The Newmark's time integration schemes is considered to solve the time-dependent equations after assembling the process and implementing boundary conditions. By using Taylor series expansions, the q t+∆t , q t+∆t andq t+∆t can be transformed into Substituting Equation (25) into Equation (24): 2 where the Newmark's parameters β 1 = 0.5 and β 2 = 0.5 are considered in this research according to the Newmark β-method.

Validation Studies
To validate the calculation method, the relative examples of Refs. [38,86] are considered by contrast. The parameters of the plate are set to 1 m in length, 1 m in width and 0.01 m in thickness. The gradient form is UD while the V c is 0.28. The parameters of the impactor are set as a mass of 0.5 kg and a radius of 0.25 m. The working conditions are a temperature of 300 K and an initial impact velocity of 3 m/s. The displacement-time curve comparative result is shown in Figure 5. It can be inferred that the results are in good agreement. The maximum displacement and contact time error could be accepted for analysis. In order to validate the equivalent layer model for the relative soft honeycomb core, a full-scale finite element simulation with an auxetic honeycomb core model was performed in contrast using the ABAQUS software, as shown in Figure 6. The sandwich structure with 0.5 mm thickness Ti-6Al-4V face sheets and auxetic honeycomb core was considered. The parameters of honeycomb core were set as: thickness h c = 23 mm; length of inclined cell rib l h = 5 mm; length of the vertical cell rib h h = 10 mm; and inclined angle θ h = −40 o . The second-order accuracy S4R elements were used to mesh the structure. Moreover, the meshes of face sheets are designed to share nodes with cores along the two interfaces, indicating the perfectly adhered to assumption. The impactor was set as an analytically rigid body ball with radius 10 mm. Furthermore, the mass was calculated according to the density 7.8 g/cm 3 . The general contact method with frictionless property was used to define the contact behavior. The initial impact velocity was 3 m/s, using predefined fields. All six degrees of freedoms of the boundary nodes were constrained to simulate clamped boundary conditions. The displacement-time curve comparative result is shown in Figure 7. It can be inferred that the results are in good agreement and the equivalent layer model could be used for the present research. To be sure, the modeling method based on continuum mechanics theory in this paper was verified. The molecular dynamic theories or nano-scale continuum modeling is a more accurate simulation method for nanomaterials such as SCNT. However, this research focuses on the qualitative study of each parameter on the structural impact response, and the continuum mechanics theory can be used to show the trend of response after verification.  Table 5 in detail. The UD form of (20/−20/20)s ply and (70/−70/70)s ply has the largest w p , smallest F c and longest t r . The FG-O form of (20/−20/20)s ply has the smallest w p , largest F c and shortest t r . While the FG-X form of (70/−70/70)s ply has the smallest w p , largest F c and shortest t r . The response of the (45/−45/45)s ply is more complicated. The UD form has the largest w p and longest t r . The FG-X form has the largest F c and shortest t r . The FG-O form has the smallest w p . The FG-V form has the smallest F c . The contact time t c of each gradient forms are nearly the same.

Volume Fractions of CNTs
The 0.11, 0.14 and 0.17 volume fractions of CNTs are considered. The surface layer of this part of the research is set as uniform distribution. The plate center displacement are shown in Figure 9. The (20/−20/20)s ply has the largest plate center displacement w p and shortest recovery time of deformation t r . The (45/−45/45)s ply has the smallest plate center displacement w p and the (70/−70/70)s ply has the longest recovery time of deformation t r . According to Table 6, the response of three stacking sequences is similar. With the volume fractions of CNTs increasing, the plate center displacement w p , recovery time of deformation t r and contact time t c decreases, while the contact force F c increases. It can be inferred that the contact stiffness increases with the volume fractions of CNTs increasing.
It is observed that increasing the stiffness of the sandwich structure by increasing the volume fraction of CNTs can lead to a reduction in the w p and an increase of the F c .

Impact Velocity
The impact velocity plays an important role in the impact response. Considering 1 m/s, 2 m/s and 3 m/s impact velocity, the plate center displacements of three stacking sequences are shown in Figure 10. The (20/−20/20)s ply has the largest plate center displacement w p and has the shortest recovery time of deformation t r . The (45/−45/45)s ply has the smallest plate center displacement w p . The (70/−70/70)s ply has the longest recovery time of deformation t r . According to Table 7, with the increased impact velocity, the plate center displacement w p and the contact force F c increased, while the recovery time of deformation t r and contact time t c decreased.
It is observed that the three stacking sequences have a slight impact on the variable ratio of w p and F c . Increasing the impact velocity from 1 m/s to 3 m/s can lead to an increase in the w p and F c by approximately 62.5% and 68%, respectively.  The low-velocity impact response of FG-CNTRC plates under various temperatures is the hotspot of its application under extreme conditions. The temperatures of 300 K, 400 K and 500 K are considered, as shown in Figure 11. Similarly to the result of various impact velocities, the (20/−20/20)s ply has the largest plate center displacement w p and has the shortest recovery time of deformation t r . The (45/−45/45)s ply has the smallest plate center displacement w p . The (70/−70/70)s ply has the longest recovery time of deformation t r . According to Table 8, with the increased temperature, the plate center displacement w p , recovery time of deformation t r and contact time t c increased, while the contact force F c decreased.
It is observed that the stiffness of the sandwich structure will reduce by increasing the temperature. From 300 K to 500 K, the w p will increase by approximately 8.4%.  The length/width ratio a/b = 0.5, 1.0 and 2.0 are considered, as shown in Figure 12. The coupling between stacking sequence and a/b makes the low-velocity impact response complicated. The a/b = 2.0 has the largest plate center displacement w p , while a/b = 0.5 is the smallest of all three stacking sequences. The responses are shown in Table 9 in detail. When a/b = 0.5, the (70/−70/70)s ply has the largest w p and smallest F c , the (45/−45/45)s ply has the smallest w p and largest F c . When a/b = 2.0, whilst the (45/−45/45)s ply has the largest w p and smallest F c , the (20/−20/20)s ply has the smallest w p and largest F c . However, the t r decreases at first and then increases with the increase in a/b. The t c increases with the increase in a/b. The results inferred that the ratio of plate length and width has a large influence on the low-velocity impact, which causes the nonlinear change phenomenon.
It is observed that the geometry scale has more influence on the impact response, due to the anisotropic honeycomb core. Using the honeycomb section as the long side of the structure can reduce the F c .

Thickness of Surface Layer
The thickness of the FG-CNTRC surface layer h s = 0.6 mm, 1.2 mm and 2.4 mm are considered, and the low-velocity impact response is shown in Figure 13. When h s = 1.2 mm and 2.4 mm, the stacking sequence has a large influence on the plate displacement w p . According to Table 10  It is observed that increasing h s can lead to a reduction in the w p and an increase in the F c by increasing the stiffness of the structure.

Conclusions
In this research, a numerical method on the low-velocity impact response of the sandwich plate with an FG-CNTRC surface and NPR honeycomb core was proposed and verified. The plate center displacement w p , recovery time of deformation t r and contact time t c increased, while the contact force F c decreased as the temperature increased. The stiffness of the structure will reduce by increasing the temperature. From 300 K to 500 K, the w p will increase by approximately 8.4%.

Conflicts of Interest:
The authors declare that the work described has not been published before; they have no conflict of interest regarding the publication of this article.